Positive-pole base change and logarithmic lifts for analytic plumbing

Put $J=e^{L(1)}(-1{)}^{L(0)}$ on V. On the two branches of a fixture define a frame indexed by $v\in V_{\le n}$ by $U_x^{-1}(x^{L(0)}v)$ and $U_y^{-1}(y^{L(0)}Jv)$. These agree on $xy=q\mathrm{\ne }0$. Indeed the centered coordinate transition has state operator $e^{x\mathrm{/}qL(1)}(-q\mathrm{/}{x}^2{)}^{L(0)}$, and applying this to $x^{L(0)}v$ gives $y^{L(0)}Jv$, using $e^{sL(1)}{a}^{L(0)}=a^{L(0)}{e}^{asL(1)}$. Declare their common frame to be free on the whole fixture. Its transition on either exterior collar is holomorphic and invertible, because x or y is nonzero there. This constructs the extension, including at $q=0$. The increasing weight filtration is preserved because positive Virasoro operators lower weight; its graded lines are ${\omega }^{-r}$. Thus $F_n$ has graded lines ${\omega }^{1-r}$, $0\le r\le n$. This also verifies the asserted dualizing generator and its sign.

On a nodal fiber let $\nu :\widehat{C}\to C$ be its normalization and Z the sum of node preimages. For a finite locally free bundle E there is an exact sequence $0\to E\to {\nu }_{\ast }{\nu }^{\ast }E\to {⨁}_{p\text{ node}}{E}_p\to 0,$ where the last arrow is the difference of the two branch values in the fixture frame. Locally this is the equality between $\mathbb{C}\{x,y\}\mathrm{/}(xy)$ and pairs of convergent one-variable series with equal constants. Choose k so that both $H^1(\widehat{C},{\nu }^{\ast }E(kD))$ and $H^1(\widehat{C},{\nu }^{\ast }E(kD)(-Z))$ vanish. The second vanishing makes evaluation at all node branches surjective, so the difference map is surjective; the displayed sequence then gives $H^1(C,E(kD))=0$. For the bundles of step 1.1, normalization line degrees are fixed combinations of $2g_v-2$, the number of node preimages, and $k\mathrm{deg}{D}_v$. Every $\mathrm{deg}{D}_v$ is positive. Line duality and the finite filtration give the two vanishings for all sufficiently large k. There are only finitely many partial-smoothing topological types, obtained by smoothing subsets of the fixed node set; their genera, marked counts and these degrees have finitely many values. Their maximum gives one $k_0$ on all P after shrinking B. This includes smooth fibers and all partial-node strata.

We prove relative base change at a point with nodes, rather than infer it from fiber dimensions. First fix the normalization parameter b and disjoint circles $\mathrm{\mid }{x}_j\mathrm{\mid }=a_j$, $\mathrm{\mid }{y}_j\mathrm{\mid }=b_j$ inside the coordinate disks. Let M be the compact normalization, let U be the union of the removed closed disks, and let $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$ be the bordered exterior. Absorb the finite incoming poles into the holomorphic bundle $H={\nu }^{\ast }{E}_0(kD)$ on M. This construction is applied only to finite bundles with filtrations by divisor line bundles; step 1.1 supplies these for the state bundles, and line bundles and finite extensions of such filtered bundles retain this property. Step 2.1 gives $H^1(M,H)=0$, so F2's filtered Green construction applies componentwise and gives a right inverse $G$ of dbar with all fixed integer Sobolev bounds. The unknowns are holomorphic H-sections on the exterior and, at each node, one common constant plus two positive Hardy series $c+{\sum }_{m\ge 1}{u}_m{x}^m+{\sum }_{m\ge 1}{v}_m{y}^m\mathrm{.}$ Fix an integer $s\ge 3$; use square-sum boundary norms with weights $(1+m{)}^{2s}{a}_j^{2m}$ and $(1+m{)}^{2s}{b}_j^{2m}$. Give the exterior space its $H^s$ trace norm, together with finitely many evaluation coordinates $\lambda$ separating the finite-dimensional section space on each component with no boundary. Here is the bounded estimate needed for completeness. In the disk frames of H, split an arbitrary boundary vector g into its nonnegative and negative Fourier parts $g_{+}$ and $g_{-}$. Their holomorphic Cauchy extensions go respectively into the disk and the exterior collar. Cut them off away from the seam, leaving them unchanged near it, to obtain the piecewise pair $(v_{\mathrm{e}\mathrm{x}\mathrm{t}},v_{\mathrm{i}\mathrm{n}})=({\chi }_{-}{g}_{-},-{\chi }_{+}{g}_{+})$. Its exterior-minus-interior jump is g. Its piecewise dbar is one global smooth form f on M: it is zero on both sides near every seam and supported away from them. Every fixed Sobolev norm of f is bounded by $C\mathrm{\parallel }g{\mathrm{\parallel }}_{H^s}$, since the Cauchy series and all their derivatives are bounded on each cutoff support a positive distance from its circle. Put $a=Gf$ and $(w_{\mathrm{e}\mathrm{x}\mathrm{t}},w_{\mathrm{i}\mathrm{n}})=(v_{\mathrm{e}\mathrm{x}\mathrm{t}}-a,v_{\mathrm{i}\mathrm{n}}-a)$. This pair is holomorphic on both sides, still has jump g, and depends linearly and boundedly on g in its boundary $H^s$ and bulk $H^s$ norms. Indeed the Cauchy Fourier extension has bulk $H^s$ norm bounded by its boundary $H^s$ norm: radial integration of a differentiated mode contributes at most a fixed constant times $(1+\mathrm{\mid }m\mathrm{\mid }{)}^{2s-1}$, dominated by the boundary weight. For a, use the higher integer Green bounds; in finite charts the elementary Fourier Cauchy--Schwarz bound from $H^{s+3}$ to $C^s$ controls its restrictions to the circles. Thus no fractional-order Green estimate is assumed. If u is an actual exterior section with trace g, the zero-jump pair $(u-w_{\mathrm{e}\mathrm{x}\mathrm{t}},-w_{\mathrm{i}\mathrm{n}})$ glues to a global holomorphic section h of H; matching traces cancel the seam term in distributional dbar, and the local scalar dbar equation makes the glued section holomorphic. On every connected component with boundary, restriction of $H^0(M,H)$ to its removed disks is injective by the identity theorem. This finite-dimensional map has a bounded inverse on its image, so $\mathrm{\parallel }h{\mathrm{\parallel }}_{H^s(M)}\le C\mathrm{\parallel }{w}_{\mathrm{i}\mathrm{n}}{\mathrm{\parallel }}_{H^s(U)}\le C^{\mathrm{\prime }}\mathrm{\parallel }g{\mathrm{\parallel }}_{H^s}$. On a component without boundary, the chosen evaluations give the corresponding bound directly. Consequently $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s(M_{\mathrm{e}\mathrm{x}\mathrm{t}})}\le C(\mathrm{\parallel }g{\mathrm{\parallel }}_{H^s(\mathrm{\partial }{M}_{\mathrm{e}\mathrm{x}\mathrm{t}})}+\mathrm{\mid }\lambda (u)\mathrm{\mid })\mathrm{.}$ To check closedness in precisely the stated trace norm, take a trace-Cauchy sequence $u_n$. The constructed $w(g_n)$ converge in bulk and boundary norms. On components with boundary, $h_n{\mathrm{\mid }}_U=-w_{\mathrm{i}\mathrm{n}}(g_n)$ converge, so the bounded finite-dimensional inverse makes $h_n$ converge as global holomorphic sections; on boundaryless components use $\lambda (u_n)$. Hence $u_n=w_{\mathrm{e}\mathrm{x}\mathrm{t}}(g_n)+h_n$ converge to an exterior holomorphic section with exactly the limiting $H^s$ trace and evaluation coordinates. The trace image is closed and the exterior space is Banach. The node Hardy spaces are Hilbert spaces, so the full unknown space is Banach as well. The matching operator $L_0$ takes exterior-minus-node traces, including the fixed invertible collar transitions from step 1.1. Its kernel is exactly $H^0(C_0,E_0(kD))$, since trace-matched holomorphic sections glue by the same distributional argument.

Here is a bounded right inverse for $L_0$. Cauchy Fourier projection splits an arbitrary $H^s$ circle jump into its inside and outside series; both projections have norm at most one in the weighted Fourier norm. Cut off the outside Cauchy section in a fixed collar, equal to that section near the circle. Its dbar is supported away from the circle, with every fixed Sobolev norm bounded by the original boundary norm, by the Cauchy formula on that smaller collar. Solve this dbar on the compact normalization using the filtered Green right inverse of F2; the first vanishing in step 2.1 makes it solvable. The resulting corrected exterior and disk sections realize the prescribed jumps. Their two node constants may differ. The evaluation-surjection in step 2.1 has a fixed finite-dimensional right inverse: prescribe all branch values by a global normalization section and add it on both its exterior and disk restrictions. This changes no jumps and corrects all constant differences. Finally choose exactly d evaluation or jet functionals whose restrictions form a basis of the dual of the d-dimensional central kernel, and subtract the unique kernel section with the same d values. Such functionals exist because a nonzero holomorphic section has a nonzero germ and the kernel is finite dimensional. This makes all d normalizing coordinates zero, rather than trying to prescribe an overdetermined set of evaluations. Every operation is bounded. Denote the resulting normalized right inverse by $R_0$. Thus $L_0$ is surjective with a bounded right inverse and finite-dimensional kernel; no assertion about compactness of a degenerating smooth fiber was used.

At q different from zero the same node normal form restricts on the x circle to $c+{\sum }_{m\ge 1}{u}_m{x}^m+{\sum }_{m\ge 1}{v}_m{q}^m{x}^{-m},$ and on the y circle to the corresponding expression with u and v interchanged. Hence $L(q)=L_0+T(q)$. For one pair the cross-substitution norm is at most ${\sup }_{m\ge 1}(\mathrm{\mid }q\mathrm{\mid }\mathrm{/}(a_j{b}_j){)}^m=\mathrm{\mid }q\mathrm{\mid }\mathrm{/}(a_j{b}_j)$ whenever that ratio is less than one; the Sobolev weights cancel. Multiplication by the fixed collar transitions costs only a fixed finite constant. With finitely many fixtures, $\mathrm{\parallel }T(q)R_0\mathrm{\parallel }<1\mathrm{/}2$ on a sufficiently small q-polydisc, and $R(q)=R_0[I+T(q)R_0{]}^{-1}$ is a convergent bounded right inverse. The normalizing evaluations remain zero. If $s_1,\dots ,s_d$ is a normalized basis of $\ker L_0$, then $s_i-R(q)L(q)s_i$ is a basis of $\ker L(q)$: the decomposition into the old kernel and the normalized complement makes L(q) invertible on that complement. This proves constant finite rank and produces the sections themselves.

The preceding construction is valid uniformly with the smooth normalization parameter b. Use the finite-ODE smooth trivialization from F2, chosen equal to the product trivialization on the finitely many coordinate disks; a partition of unity makes its horizontal lift have this property. The compact-normalization Green inverses and the finite node-evaluation inverse vary smoothly after shrinking B, by F2's Neumann construction and a nonzero evaluation minor. Thus the bounds in the matching and Neumann constructions are uniform. The constructed kernel sections are smooth in b and holomorphic in q. They are in fact holomorphic in b as well: choose the normalizing evaluations at finitely many holomorphic auxiliary sections, with holomorphic bundle frames and finite vertical jets. The canonical base $(0,1)$ derivative of a fiber-holomorphic matched section is again fiber holomorphic and matched, since total bundle transitions are holomorphic and total dbar squares to zero. Applied to one of the normalized kernel sections, all its normalizing evaluations are zero. The uniqueness of the normalized matching solution therefore makes this derivative zero. The same argument proves holomorphic parameter dependence for the normalized inverse on holomorphic jump data. This is the finite-kernel argument of F2, now with matching boundary conditions; it does not assume a holomorphic trivialization of the varying normalization curves.

The normal form produces actual holomorphic sections on the total node, not just separate fiber sections. The positive-series coefficients depend holomorphically on $(q,b)$ and have the uniform Hardy bounds of step 5.1. On smaller disks their two series converge normally to $c(q,b)+{\sum }_{m\ge 1}{u}_m(q,b)x^m+{\sum }_{m\ge 1}{v}_m(q,b)y^m\mathrm{.}$ Restricting to $xy=q$ gives the solved traces. Conversely any total holomorphic section has this normal form, by replacing each mixed monomial $x^a{y}^b$ by $q^{\min (a,b)}{x}^{a-b}$ or its y counterpart; Cauchy estimates on smaller disks justify regrouping. The kernel basis is therefore a basis of actual sections on every fiber and its holomorphic coefficient functions describe every total section. Evaluation/jet normalization shows those coefficients are holomorphic. Pulling this basis and its normalization back along any holomorphic manifold map proves the section-level base-change assertion. At a point with some nonzero sewing parameters absorb those already smooth fixtures into the smooth normalization family and apply the same argument to the remaining zero parameters. F2 treats points with no nodes. These local descriptions glue canonically on all P.

Apply the preceding fiber-vanishing and base-change construction to all sufficiently positive twists of each stated bundle. The principal-part arrow has kernel the smaller-pole bundle, and its fiber cokernel is zero by the first cohomology vanishing. Its holomorphic matrix thus has a local maximal-minor right inverse. This proves the sheaf exact sequence $0\to {\pi }_{\ast }{F}_n(kD)\to {\pi }_{\ast }{F}_n(\mathrm{\ell }D)\to P_{k,\mathrm{\ell }}\to 0.$ F4, on the entire P, makes it surjective on global sections. Crucially its kernel is the same fixed finite bundle for every larger $\mathrm{\ell }$; there is no intersection of infinitely many shrinking base neighborhoods.

Let $\mathrm{\Delta }$ be the coordinate divisor in P, and let $\mathcal{L}$ be the locally free sheaf of logarithmic vector fields on $\mathcal{C}$ whose image lies in $T_P(-\log \mathrm{\Delta })$. Away from the nodes it is the ordinary tangent bundle with logarithmic base directions. In a node chart $xy=q_j$, the node directions are generated in $\mathcal{L}$ by $x{\mathrm{\partial }}_x$ and $y{\mathrm{\partial }}_y$; the other base directions are generated by $q_i{\mathrm{\partial }}_{q_i}$ and the $b$-coordinate fields. The map to the base sends both node generators to $q_j{\mathrm{\partial }}_{q_j}$, so its kernel is generated by $x{\mathrm{\partial }}_x-y{\mathrm{\partial }}_y$, dual to $dx\mathrm{/}x$. These local frames give the exact sequence of finite bundles $0⟶{\omega }^{-1}(kD)⟶\mathcal{L}(kD)⟶{\pi }^{\ast }{T}_P(-\log \mathrm{\Delta })(kD)⟶0.$ For sufficiently large $k$, the fiber $H^1$ of ${\omega }^{-1}(kD)$ vanishes by the normalization-degree argument of step 2.1, which applies to the stated fixed integral powers of $\omega$. The quotient is a finite direct sum of ${\mathcal{O}}_{C_b}(k{D}_b)$ because $T_P(-\log \mathrm{\Delta })$ is trivial on the coordinate polydisc P; its fiber $H^1$ also vanishes, since D meets every component of each normalization. The long exact sequence therefore gives $H^1(C_b,{\mathcal{L}}_b(k{D}_b))=0$. The finite matching construction of steps 3.1–7.1 applies to these finite bundles and their displayed line filtrations, so their direct images are finite locally free and commute with base change. Pushing forward the exact sequence gives $0⟶{\pi }_{\ast }{\omega }^{-1}(kD)⟶{\pi }_{\ast }\mathcal{L}(kD)⟶T_P(-\log \mathrm{\Delta })\otimes {\pi }_{\ast }{\mathcal{O}}_{\mathcal{C}}(kD)⟶0.$ The constant meromorphic function 1 is a holomorphic section of ${\mathcal{O}}_{\mathcal{C}}(kD)$, and F4 gives $H^1(P,{\pi }_{\ast }{\omega }^{-1}(kD))=0$. Hence the map on global sections is surjective. Lift each coordinate logarithmic generator to a global section of $\mathcal{L}(kD)$; the resulting vector field has poles only on D and projects to the specified base generator. Its coefficients are holomorphic at each fixture because D is disjoint from the fixtures. On its own jth fixture, $q_j{\mathrm{\partial }}_{q_j}$ is the image of $a_j{x}_j{\mathrm{\partial }}_{x_j}+b_j{y}_j{\mathrm{\partial }}_{y_j}$, so $a_j+b_j=1$. On every other fixture the base component is zero in that fixture's parameter direction, so the corresponding coefficients sum to zero. This proves the asserted lifts without a local-to-global gluing assumption.

Choose the finite homogeneous state representatives A and finite incoming generators in F3, and let n bound their state weights. Use step 8.1 with one fixed $k_0=k_0(n)$. For every arbitrarily large $\mathrm{\ell }$ it lifts the principal part $a{t}_i^{-\mathrm{\ell }}d{t}_i$ at one incoming section with zero higher principal parts at all others, leaving error poles of order at most $k_0$. The strict-index spanning reduction of F3 writes every sufficiently high auxiliary-degree vector as a sum $a_{(-\mathrm{\ell })}^{(i)}{w}^{\circ }$ with $\mathrm{\ell }>n+k_0$. The lifted current relation replaces this by vectors of strictly smaller degree: all remainder modes have degree at most $n+k_0-1$. Induction reduces every W vector to a fixed finite auxiliary-weight window G, with coefficients holomorphic on the whole P. Thus $G\otimes \mathcal{O}(P)$ surjects modulo the global Ward span. The same construction works on open subpolydiscs. Only fixed finite current bundles and their actual sections have been used; no coherence of the infinite Ward quotient or nodal local freeness is needed.